Use and Abuse of the Fisher Information Matrix in the Assessment of Gravitational-Wave Parameter-Estimation Prospects

TL;DR

The paper critically examines the limitations of the Fisher matrix in gravitational-wave parameter estimation, highlighting common pitfalls, and offers practical methods to improve its reliability, especially at low SNR or with weak parameter dependence.

Contribution

It identifies key issues with Fisher matrix predictions in gravitational-wave analysis and provides guidelines and formulas to address these limitations effectively.

Findings

Fisher matrices can be singular or nearly singular, affecting parameter estimation.

Prior information is crucial when Fisher matrix predictions are unreliable.

Higher-order corrections improve accuracy of parameter estimates at moderate SNR.

Abstract

The Fisher-matrix formalism is used routinely in the literature on gravitational-wave detection to characterize the parameter-estimation performance of gravitational-wave measurements, given parametrized models of the waveforms, and assuming detector noise of known colored Gaussian distribution. Unfortunately, the Fisher matrix can be a poor predictor of the amount of information obtained from typical observations, especially for waveforms with several parameters and relatively low expected signal-to-noise ratios (SNR), or for waveforms depending weakly on one or more parameters, when their priors are not taken into proper consideration. In this paper I discuss these pitfalls; show how they occur, even for relatively strong signals, with a commonly used template family for binary-inspiral waveforms; and describe practical recipes to recognize them and cope with them. Specifically, I answer the following questions: (i) What is the significance of (quasi-)singular Fisher matrices, and how must we deal with them? (ii) When is it necessary to take into account prior probability distributions for the source parameters? (iii) When is the signal-to-noise ratio high enough to believe the Fisher-matrix result? In addition, I provide general expressions for the higher-order, beyond--Fisher-matrix terms in the 1/SNR expansions for the expected parameter accuracies.